By Korner T.A.

Those are the skeleton notes of an undergraduate direction given on the PCMI convention in 2003. I should still wish to thank the organisers and my viewers for an exceptionally stress-free 3 weeks. The record is written in LATEX2e and will be to be had in tex, playstation , pdf and clvi layout from my domestic web page

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If f and fn lie in S we say that fn → f if, for each fixed pair of positive 2 m integers r and m, we have (1 + x ) S (r) (fn (x) − f (r) (x)) → 0 uniformly on R. It turns out that the Schwartz space is beautifully adapted to the Fourier transform. 2. If f ∈ S, let us write Ff (λ) = fˆ(λ) = ∞ f (t)e−iλt dt. −∞ Then Ff is a well defined element of S. The map F : S → S is linear and F 2 = 2πJ where Jf (x) = f (−x). Thus F : S → S is a bijection. Further F is continuous in the sense that fn → f implies Ffn → Ff .

Dym and H. P. McKean Fourier Series and Integrals Academic Press, 1972. [2] F. G. Friedlander Introduction to the Theory of Distributions CUP, 1982. [There is a second edition also published by CUP in 1998 with an additional chapter by M. ] [3] H. Helson Harmonic Analysis Adison–Wesley, 1983. [4] Y. Katznelson An Introduction to Harmonic Analysis Wiley, 1963. ] [5] T. W. K¨orner Fourier Analysis CUP, 1988. 19 Exercises Here are some exercises. They are at various levels and you are not expected to do all of them.

Iii) Show that given a < α < β < b we can find an infinitely differentiable function f : R → R with 1 ≥ f (x) ≥ 0 for all x, f (x) = 1 for all x ∈ [α, β], f (x) > 0 for x ∈ (a, b) and f (x) = 0 for all x ∈ / [a, b]. 32. ) (i) Let T ∈ D . We say that an open interval (a, b) ∈ A if we can find an η > 0 such that, if f ∈ D and f (x) = 0 whenever x ∈ / (a − η, b + η) then T, f = 0. Let U = (a,b)∈A (a, b) and supp T = T \ U . Explain why supp T is closed. 12, that if K is closed set with K ∩ supp T = ∅, f ∈ D and f (x) = 0 for all x ∈ / K, then T, f = 0.